Growing plants under lights works very well. I started with fluorescent lights for the orchids but now also use them for my tropical flowering/fruiting plants, geraniums, jasmines, citrus, figs, passiflora, etc. One year, I grew saffron and strawberries. Even full sun plants produced flowers and fruit.
We had another member who had a very nice closet set up so it can be successfully done. With LED lights instead of fluorescent lights, you won't need to worry as much about the heat.
I have no idea how many plants/orchids you are planning to grow in your closet and what your budget is but here are some possible items that might be of use:
Here is an example of LED lights:
Technical Data - SunBlaster Lighting
They have different sizes. How many lights you need per shelf will depend on what you grow. LED lights wouldn't get as hot as the fluorescent lights that I use. I use all fluorescent lights except for one two foot LED strip. If you have a hydroponics store, they can help you select the lights and ballasts. Make certain that the ballasts come with a way to hang them (mine has holes I can run the S-hooks through) and come from a reliable brand for safety.
These are great shelves:
Amazon.com: Alera ALESW604818BA Complete Wire Shelving Unit w/Caster, Four-Shelf, 48 x 18 x 72, Black Anthracite: Home & Kitchen
I attach my lights snug against the shelf above to give my plants the maximum space to grow. I use S hooks for the big four-foot, four lamp ballasts.
I would enclose all but the front of your shelves with 6 mil. plastic sheeting to prevent mold in the closet (including underneath). In addition, I line each shelf with $5 clear plastic table clothes that I cut to size to protect the lights on the shelf below. Between the shelves and the sheeting, you could even place reflective material to help distribute the light.
Using a timer makes it all so much easier. I have had this one about four years:
Century 7 Day Heavy Duty Digital Programmable Timer - Dual Outlet (Single Control) - - Amazon.com
I hate the directions for programming it but better directions can be found online.
Last of all, use a power strip that has a fuse and surge protection to protect your lights during power surges.
Pictured is a two-foot, single-bulb fluorescent light tucked under a shelf in a dark spot. The orchids and geraniums bloomed.